1. Inference on transformed variables
Note that when performing inferential analysis on transformed variables, an important aspect to keep in mind is the interpretation of the slope coefficients used in the transformed models.
2. Interpreting coefficients - linear
Recall that in a simple linear model setting, the true population slope describes the change in expected response when X is one unit larger. That is, for every additional unit of X, we expect that the response variable, Y, is beta-1 units larger, on average.
3. Interpreting coefficients - nonlinear X
When X is non-linear, the slope coefficient gives the additional expected increase in non-linear units. Here, when the log of X goes up by one unit, the expected value of Y is beta-1 units larger. This interpretation would work for any function of X. That is, the slope represents the change average expected Y units for a one unit change in whatever function of X you are working with.
4. Interpreting coefficients - nonlinear Y
Similarly, when Y is non linear, you can interpret the slope coefficient by saying that a one unit increase in X provides an expected increase of beta-1 in log-Y units. The interpretation will again work for any functional transformation of Y in that the slope coefficient represents an average expected change in the new units given a one unit change in X.
5. Interpreting coefficients - both nonlinear
When the variables have been transformed so that both are on a non-linear scale, you can again model your interpretation based on the units at hand:
a one-unit increase in log-X is associated with an expected increase of beta-1 in log-Y.
And again, the interpretation in transformed units will hold for any functional transformation of X and Y.
6. Interpreting coefficients - both natural log (special case)
However, when both variables are transformed using natural log (as they are here), there is a special interpretation that usually holds. The derivation of the interpretation is beyond the scope of this class, but often in a natural log-log situation, the beta coefficient can be interpreted as the percent change in Y for each 1-percent change in X.
7. Let's practice!
Note that with the housing data, it is most appropriate to run a model with log transformed variables (so that the technical conditions hold). So now it's your turn to interpret the new transformed model.